Moving Finite Unit Tight Frames for S

Frames for R can be thought of as redundant or linearly dependent coordinate systems, and have important applications in such areas as signal processing, data compression, and sampling theory. The word “frame” has a different meaning in the context of differential geometry and topology. A moving frame for the tangent bundle of a smooth manifold is a basis for the tangent space at each point which varies smoothly over the manifold. It is well known that the only spheres with a moving basis for their tangent bundle are S, S, and S. On the other hand, after combining the two separate meanings of the word “frame”, we show that the n-dimensional sphere, S, has a moving finite unit tight frame for its tangent bundle if and only if n is odd. We give a procedure for creating vector fields on S2n−1 for all n ∈ N, and we characterize exactly when sets of such vector fields form a moving finite unit tight frame on S2n−1. This gives as well a new method for constructing finite unit tight frames for Hilbert spaces.